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Visual Complex Functions An Introduction with Phase Portraits

Visual Complex Functions An Introduction with Phase Portraits This book provides a systematic introduction to functions of one complex variable. Its novel feature is the consistent use of special color representations – so-called phase portraits – which visualize functions as images on their domains. Reading Visual Complex Functions requires no prerequisites except some basic knowledge of real calculus and plane geometry. The text is self-contained and covers all the main topics usually treated in a first course on complex analysis. With separate chapters on various construction principles, conformal mappings and Riemann surfaces it goes somewhat beyond a standard programme and leads the reader to more advanced themes. In a second storyline, running parallel to the course outlined above, one learns how properties of complex functions are reflected in and can be read off from phase portraits. The book contains more than 200 of these pictorial representations which endow individual faces to analytic functions. Phase portraits enhance the intuitive understanding of concepts in complex analysis and are expected to be useful tools for anybody working with special functions – even experienced researchers may be inspired by the pictures to new and challenging questions. Visual Complex Functions may also serve as a companion to other texts or as a reference work for advanced readers who wish to know more about phase portraits. ; This book introduces the basic concepts of complex analysis. It details how properties of a function are reflected in and can be recovered from its phase portrait. The book features color illustrations which reveal the beauty of complex functions.; Preface.- 1. Getting Acquainted.- 2. Complex Functions.- 3. Analytic Functions.- 4. Complex Calculus.- 5. Construction Principles.- 6. Conformal Mappings.- 7. Riemann Surfaces ; From the reviews: “This textbook is an introduction to the classical theory of functions of one complex variable. Its distinctive feature are the graphical representations of functions, being the most useful tool in teaching and generally in mathematics. … The self-sufficiency of the textbook and the broad range of graphical examples makes the book useful for students as well as teachers of mathematics. … the book can be warmly recommended both to experts and to a new generation of mathematicians.” (Stanislawa Kanas, Zentralblatt MATH, Vol. 1264, 2013) “Anyone who works with complex variables should read this book. … Visual Complex Functions is a beautiful and careful presentation of an entire advanced introduction to complex analysis based on phase portraits and, where appropriate, other kinds of computer-generated pictures. … My understanding of many ideas and phenomena deepened through reading this book.” (Lloyd N. Trefethen, SIAM Review, Vol. 55 (4), 2013) ; Elias Wegert is a Professor of Mathematics at the Technische Universität Bergakademie Freiberg. His main research interests are in Complex and Nonlinear Analysis; he is the author of a book on nonlinear Riemann-Hilbert problems and their applications. Wegert has experience in mathematical teaching at different levels for about 30 years. He is also committed to mathematical competitions for high school students.; This book provides a systematic introduction to functions of one complex variable. Its novel feature is the consistent use of special color representations – so-called phase portraits – which visualize functions as images on their domains. Reading Visual Complex Functions requires no prerequisites except some basic knowledge of real calculus and plane geometry. The text is self-contained and covers all the main topics usually treated in a first course on complex analysis. With separate chapters on various construction principles, conformal mappings and Riemann surfaces it goes somewhat beyond a standard programme and leads the reader to more advanced themes. In a second storyline, running parallel to the course outlined above, one learns how properties of complex functions are reflected in and can be read off from phase portraits. The book contains more than 200 of these pictorial representations which endow individual faces to analytic functions. Phase portraits enhance the intuitive understanding of concepts in complex analysis and are expected to be useful tools for anybody working with special functions – even experienced researchers may be inspired by the pictures to new and challenging questions. Visual Complex Functions may also serve as a companion to other texts or as a reference work for advanced readers who wish to know more about phase portraits. ; First introduction to complex functions systematically using phase portraits (which are expected to become an indispensable tool for exploring complex functions in teaching, mathematical research and applications alike) Reorganization of the traditional material covered by textbooks in complex analysis and emphasis on special aspects of the topic. A companion to rather than a substitute for existing textbooks on complex analysis, although mostly self-contained Enhances intuitive understanding of basic concepts and special functions by visualizing complex functions by phase portraits Complete text equipped with color illustrations ; DE http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

Visual Complex Functions An Introduction with Phase Portraits

374 pages

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Publisher
Springer Basel
Copyright
Copyright © Springer Basel AG
DOI
10.1007/978-3-0348-0180-5
Publisher site
See Book on Publisher Site

Abstract

This book provides a systematic introduction to functions of one complex variable. Its novel feature is the consistent use of special color representations – so-called phase portraits – which visualize functions as images on their domains. Reading Visual Complex Functions requires no prerequisites except some basic knowledge of real calculus and plane geometry. The text is self-contained and covers all the main topics usually treated in a first course on complex analysis. With separate chapters on various construction principles, conformal mappings and Riemann surfaces it goes somewhat beyond a standard programme and leads the reader to more advanced themes. In a second storyline, running parallel to the course outlined above, one learns how properties of complex functions are reflected in and can be read off from phase portraits. The book contains more than 200 of these pictorial representations which endow individual faces to analytic functions. Phase portraits enhance the intuitive understanding of concepts in complex analysis and are expected to be useful tools for anybody working with special functions – even experienced researchers may be inspired by the pictures to new and challenging questions. Visual Complex Functions may also serve as a companion to other texts or as a reference work for advanced readers who wish to know more about phase portraits. ; This book introduces the basic concepts of complex analysis. It details how properties of a function are reflected in and can be recovered from its phase portrait. The book features color illustrations which reveal the beauty of complex functions.; Preface.- 1. Getting Acquainted.- 2. Complex Functions.- 3. Analytic Functions.- 4. Complex Calculus.- 5. Construction Principles.- 6. Conformal Mappings.- 7. Riemann Surfaces ; From the reviews: “This textbook is an introduction to the classical theory of functions of one complex variable. Its distinctive feature are the graphical representations of functions, being the most useful tool in teaching and generally in mathematics. … The self-sufficiency of the textbook and the broad range of graphical examples makes the book useful for students as well as teachers of mathematics. … the book can be warmly recommended both to experts and to a new generation of mathematicians.” (Stanislawa Kanas, Zentralblatt MATH, Vol. 1264, 2013) “Anyone who works with complex variables should read this book. … Visual Complex Functions is a beautiful and careful presentation of an entire advanced introduction to complex analysis based on phase portraits and, where appropriate, other kinds of computer-generated pictures. … My understanding of many ideas and phenomena deepened through reading this book.” (Lloyd N. Trefethen, SIAM Review, Vol. 55 (4), 2013) ; Elias Wegert is a Professor of Mathematics at the Technische Universität Bergakademie Freiberg. His main research interests are in Complex and Nonlinear Analysis; he is the author of a book on nonlinear Riemann-Hilbert problems and their applications. Wegert has experience in mathematical teaching at different levels for about 30 years. He is also committed to mathematical competitions for high school students.; This book provides a systematic introduction to functions of one complex variable. Its novel feature is the consistent use of special color representations – so-called phase portraits – which visualize functions as images on their domains. Reading Visual Complex Functions requires no prerequisites except some basic knowledge of real calculus and plane geometry. The text is self-contained and covers all the main topics usually treated in a first course on complex analysis. With separate chapters on various construction principles, conformal mappings and Riemann surfaces it goes somewhat beyond a standard programme and leads the reader to more advanced themes. In a second storyline, running parallel to the course outlined above, one learns how properties of complex functions are reflected in and can be read off from phase portraits. The book contains more than 200 of these pictorial representations which endow individual faces to analytic functions. Phase portraits enhance the intuitive understanding of concepts in complex analysis and are expected to be useful tools for anybody working with special functions – even experienced researchers may be inspired by the pictures to new and challenging questions. Visual Complex Functions may also serve as a companion to other texts or as a reference work for advanced readers who wish to know more about phase portraits. ; First introduction to complex functions systematically using phase portraits (which are expected to become an indispensable tool for exploring complex functions in teaching, mathematical research and applications alike) Reorganization of the traditional material covered by textbooks in complex analysis and emphasis on special aspects of the topic. A companion to rather than a substitute for existing textbooks on complex analysis, although mostly self-contained Enhances intuitive understanding of basic concepts and special functions by visualizing complex functions by phase portraits Complete text equipped with color illustrations ; DE

Published: Aug 30, 2012

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